Frobenius-Schur indicators for subgroups and the Drinfel’d double of Weyl groups

Abstract

Abstract. If G is any finite group and k is a field, there is a natural construction of a Hopf algebra over k associated to G, the Drinfel'd double D(G). We prove that if G is any finite real reflection group, with Drinfel'd double D(G) over an algebraically closed field k of characteristic not 2, then every simple D(G)-module has Frobenius-Schur indicator +1. This generalizes the classical results for modules over the group itself. We also prove some new results about Weyl groups. In particular, we prove that any abelian subgroup is inverted by some involution. Also, if E is any elementary abelian 2-subgroup of the Weyl group W , then all representations of C W (E) are defined over Q.

Involution models of finite Coxeter groups

“…It is noted in [7] that Theorem 2 has been verified with a computer, along with more subgroups of H 4 , and in fact the irreducible characters of the centralizers of involutions in H 4 are all orthogonal.…”

“…Given the application to the case of H 4 in Proposition 3, it may be of interest to investigate the reality properties and Schur indices of the normalizers of parabolic subgroups of finite Coxeter groups in general. This investigation has begun in the paper of Guralnick and Montgomery [7], with the study of Frobenius-Schur indicators of certain subgroups of Weyl groups.…”

Representations of Some Hopf Algebras Associated to the Symmetric Group S n

“…Recently it was shown in [5] that D(G), the Drinfel'd double of a finite group, is totally orthogonal for any finite real reflection group G.…”

Higher Indicators for the Doubles of Some Totally Orthogonal Groups

“…It has been conjectured that non-negativity of the indicators for D(G) holds provided we further require that G be a real reflection group. Such groups are well-known to satisfy conditions i), ii), and iv) of our theorem (the fifth part remains unproven, except for the dihedral [13] and symmetric [23] groups), and in [7] it was shown that D(G) is also totally orthogonal. Theorem 7.5 shows that the condition of being generated by reflections in the conjecture cannot be relaxed to simply "generated by involutions".…”